Signatures on semilocal rings

Manfred Knebusch, Alex Rosenberg, Roger Ware
1972 Bulletin of the American Mathematical Society  
We announce extensions of a part of the Artin-Schreier theory of real fields to semilocal rings. Detailed proofs will appear elsewhere. A always denotes a (not necessarily noetherian) semilocal commutative ring such that no residue class field has only two elements. A signature on A is a homomorphism a from the unit group, A*, of A to {± 1} with a{-1) = -1 and a(l 2 + am 2 ) = 1 for all triples (a, /, m) in A* x A x A such that I 2 + am 2 is a unit and cr(a) = 1. EXAMPLES, (i) If A is an
more » ... l domain and < is a total ordering of A then a :A* -» {± 1} defined by a(a) = 1 if a > 0 and a(a) = -1 if a < 0 is a signature. If A is a field the signatures correspond bijectively with the orderings of A. (ii) Let A be the local ring of the affine curve X 2 + Y 2 = 0 over the real field R at (0,0). Then the signature <T:A* 4J?*^{±1} obtained by composing the evaluation map v at (0,0) and the unique signature s of R does not arise from an ordering of A. (iii) For valuation rings we are able to analyze the situation to some extent : Let A be a valuation ring with maximal ideal m. Then any signature G arises from an ordering of A, \iA has rank one and o(\ + m) ^ 1, then a arises from a unique ordering. If A is a discrete rank one valuation ring and (7(1 + m) = 1, then there are exactly two orderings on A inducing the signature a. Let A now again be a general semilocal ring as above. PROPOSITION 1. Let a be a signature, a l9 ...,a r units of A, and l u ...,l r elements of A such that b -l\a x + • • • + l 2 a r is also a unit. Then G{a^) = • • • = o(a r ) = 1 implies <r(b) = 1. DEFINITION. A subset M oîA* is saturated if M is a subgroup of A* and a u ..., a r in M implies b in M for all units b = l\a x + • • • + l 2 a r with /,-in A. Thus Proposition 1 states that for any signature a the set r(<r)= {a in y4*| <r(a) = 1} is saturated. AMS 1970 5t/tyecr classifications. Primary 15A63, 13H99; Secondary 06A70, 12J15.
doi:10.1090/s0002-9904-1972-12852-0 fatcat:rq5irfe63fejth4wpthil5l6ea